Bounds on the 3-rainbow Domination

Let \(G(V, E)\) be a simple graph. For a labeling \(\partial: V \cup E \to \{1, 2, 3, \dots, k\}\), the weight of a vertex \(x\) is defined as \(wt(x) = \partial(x) + \sum\limits_{xy \in E} \partial(xy)\). \(\partial\) is called a vertex irregular total \(k\)-labeling if for every pair of distinct vertices \(x\) and \(y\), \(wt(x) \neq wt(y)\). The minimum \(k\) for which the graph \(G\) has a vertex irregular total \(k\)-labeling is called the total vertex irregularity strength of \(G\) and is denoted by \(tvs(G)\). In this paper, we obtain a bound for the total vertex irregularity strength of honeycomb and honeycomb derived networks.